Improvement of the SPH Method for Pin-by-Pin Core Calculations

In this paper, improvement of the SPH method (the improved SPH method) is proposed. The SPH method is commonly used in pin-by-pin mesh core calculations to reduce cell-homogenization error. The investigation revealed that the normalization condition of the SPH factor in the conventional SPH method is not appropriate for multi-assembly calculations in which different assembly types are adjacent. Since the conventional normalization condition does not incorporate flux discontinuity between assemblies, cell homogenization error in assembly peripheral region becomes larger. In the improved SPH method, the SPH factor is divided by an averaged “cell-level” discontinuity factor obtained in each fuel assembly. Though the SPH factor is somewhat modified from the conventional value, no additional homogenization parameters (e.g. discontinuity factor) is necessary in core calculations. Test calculations were carried out in a simplified one-dimensional slab and two-dimensional PWR colorset geometries that simulate part of actual core geometry. The calculation results showed that the improved SPH method effectively reduce the cell-level homogenization error especially in assembly peripheral region. Since we can easily implement the improved SPH method by slight code modifications, it can be a promising candidate of the cell-homogenization method for pin-by-pin core calculations.     

Generalized Bias Factor Method for Accurate Prediction of Neutronics Characteristics

A generalized bias factor method is proposed to improve the prediction accuracy of neutronics characteristics of a target core. The generalized bias factor method uses conventional bias factors calculated for several critical assemblies. The weighting factors for individual bias factors are determined to minimize the variance of neutronic characteristics of the target core. Numerical calculations are performed to investigate the uncertainty reductions of neutronics characteristics for a tight-lattice core. Though the uncertainty is not remarkably reduced for k_eff , that for the reaction rate ratio of ²³⁸U capture/²³⁹Pu fission is remarkably reduced: For example, the uncertainty reduction of the reaction rate ratio in the upper core is 0.871 for the present method, and 0.657 for the conventional bias factor method.     

Development of a hybrid method to improve the sensitivity and uncertainty analysis for homogenized few-group cross sections

In the framework of two-step method of reactor core calculation, few-group homogenized cross sections generated by lattice-physics calculations are key input parameters for the three-dimensional full-core calculation. Conventional method for few-group cross-sections sensitivity and uncertainty (S&U) analysis related to the nuclear data was performed based on the effective self-shielding cross sections instead of the continuous-energy cross sections, which means resonance self-shielding effect (implicit effect) is neglected. Furthermore, the multi-group covariance data is generated from the continuous-energy cross sections. Therefore, in order to perform S&U analysis with respect to the continuous-energy cross sections for both accuracy and consistency, a hybrid method is proposed in this paper. The subgroup-parameter sensitivity-coefficients are calculated based on the direct perturbation (DP) method. The sensitivity-coefficients of the effective self-shielding cross sections and the responses (k_eff and few-group homogenized cross sections) are calculated based on the generalized perturbation theory (GPT). A boiling water reactor (BWR) pin-cell problem under different power conditions is calculated and analyzed. The numerical results reveal that the proposed hybrid method improves the sensitivity-coefficients of eigenvalue and few-group homogenized cross sections. The temperature effects on the sensitivity-coefficients are demonstrated and the uncertainties are analyzed.     

Measurement and analysis of feedback reactivity in the Monju restart core

The purpose of this study is to develop a feedback reactivity measurement technique in the Japanese prototype fast breeder reactor Monju and to validate calculation methodology to forecast the nuclear feedback phenomena. A feedback reactivity measurement technique has been developed based on a reactivity model featuring components that depend on the reactivity coefficients, denoted as reactor power (K_R) and reactor vessel inlet temperature (K_in). This technique can precisely measure the two reactivity components simultaneously and was applied to the feedback reactivity experiment conducted in the Monju system start-up test in May 2010. A thorough evaluation considering all possible biases and uncertainties demonstrated that the reactivity coefficients can be evaluated with a measurement uncertainty smaller than 3%. The evaluated reactivity coefficients were simulated considering the temperature distribution in the core. The calculated and measured values of K_R agreed within 1%, and the value of K_in was consistent with that reported in a previous isothermal temperature coefficient experiment. The measured and calculated fuel subassembly outlet temperatures also agreed well within 0.2°C, which supports the validity of the temperature calculation.     

Buckling Correction for Experimental Power Distribution within Fuel Assemblies in a Finite System

For design calculations to determine the local power distribution in a fuel assembly of a power reactor, the neutron flux is usually assumed to be symmetrical at the outer boundary of the assembly. In actuality, experimental data on power distributions are obtained in a finite system where this symmetry does not apply, so that the calculated values cannot be directly compared with observed data. In a zero power critical experiment in particular, the measurement must be performed in a fairly small core assembly so economize the amount of fuel materials to be used for simulation. This introduces the necessity of special considerations in the comparison between design and observed data.

The authors propose a method incorporating direct corrections to the experimentally determined. power distributions based on the geometrical buckling of the system. In this method the experimental power P ⁰ (r) is divided by the neutron flux Q (r) which is determined in the critical state with geometrical buckling in a bare (one neutron energy group) reactor, neglecting the reflector region of the experimental system. A sample application of the method to an actual light water lattice has confirmed the validity of the method.     

Parallelization of Monte Carlo Code MCACE for Shielding Analysis and Estimation of Its Efficiency by Simulator

An improved coarse-mesh discrete ordinates method has been developed for three-dimensional hexagonal transport calculations of high-conversion light water reactors and fast reactors. This method employs a new weighted diamond difference approximation which is obtained by using the neutron balance equations in divided submeshes. The weight is a function of neutron direction and scaler flux, and this method can be easily incorporated into conventional discrete ordinates transport codes.

The present method was applied to hexagonal fuel assembly calculations of high-conversion reactor and fast reactor core calculations, and the results were compared with those of Monte- Carlo calculations. The values of kefi and power distributions agreed with each other within 0.5 and 3%, respectively, verifying accuracy of the present improved coarse-mesh discrete ordinates transport calculation method.     

Meeting fuel temperature limits in an HTR-Module Reactor during depressurized core heat-up

The thermal behaviour of an HTR-Module Reactor is discussed for the design basis event of core heat-up after fast depressurization taking into account the most unfavourable initial state and uncertainties of input data. The reactor is designed to retain its fission products inside the fuel coatings even if all active components for decay heat removal and reactivity control should fail. To meet this goal maximum fuel temperatures during core heat-up should not exceed the technological limit of 1620°C, for which the integrity of the fuel coatings has been proven experimentally.Two-dimensional thermal-hydraulic calculations show that the maximum fuel temperature during core heat-up is expected to be 1472°C taking into account nominal full power operation as an initial state, a sudden depressurization in the beginning of the event, and nominal input data. The most unfavourable initial state is the steady state operation close to the scram set points, i.e. 105% power and increased cold and hot gas temperatures. Accounting for this leads to a maximum fuel temperature of 1522°C. Relevant uncertainties of input data are those of decay heat production, power distribution and core thermal conductivity and specific heat capacity. Their individual standard deviations can be combined to an integral uncertainty margin of ±86 K which covers two standard deviations. Hence the maximum fuel temperature taking into account unfavourable initial state and uncertainties is 1608°C.     

多群核数据不确定性对堆芯物理计算的影响

核数据不确定性是造成反应堆物理计算结果不确定性的重要因素之一。基于所需抽样核数据的协方差矩阵开发了随机抽样模块(Stochastic Sampling,SAMP),在此基础上利用SCALE(Standardized Computer Analyses for Licensing Evaluation)软件包实现了混合法和随机抽样法两种不确定性分析方法,以研究多群核数据不确定性对堆芯物理计算的影响。以3×3假想堆芯为对象,对两种方法进行了验证,然后应用于国际原子能机构(International Atomic Energy Agency,IAEA)燃料管理基准题中的Almaraz核电厂首循环堆芯。分析结果表明,两种方法结果符合良好,Almaraz核电厂堆芯keff不确定性约为0.5%,堆芯径向和轴向功率的最大不确定性分别为1.9%和0.45%。     

Effect of Geometrical Approximation on Multiplication Factor in Small Fast Reactor Core

In most of the calculations using analytical methods a reactor core is approximated as cylinder and the reactor parameters are calculated using two-dimensional computer codes. While such calculations are useful in scoping studies in view of azimuthal asymmetry in the actual reactor core these calculations could entail errors of unknown magnitude. The present study reports our estimate of such errors in K _eff with the instance of fast reactor having 22 and 23 fuel subassemblies. The K _eff are calculated using Monte Carlo code KENO and Hansen-Roach cross section set, modelling the core in two different ways, (1) by approximating the core to a cylinder (2-D calculation), (2) by near exact representation of the core (3-D calculation). The difference in K _eff is appreciable between 2-D and 3-D calculations.

Experimental values are adduced in support of these calculations.     

Method of Experimental Power Reactor Noise Analysis without Use of High Power Reactor

A method is proposed for conducting power reactor noise analysis without recourse to an actual high power reactor. The basic concept is to simulate the power reactor noise by integrating the different elements constituting the actual reactor noise, such as the random noise-generating force, the zero-power reactor transfer function, and feed- back loops between the reactor power and the noise-generating force.

For the simulation study, a nucleate boiling noise generator and a single feed-back loop were divised to permit experiments with flux-related heater input using the fast neutron source reactor YAYOI of the University of Tokyo.

This report discusses problems encountered in applying the proposed method to the simulation of power reactor noise, and presents some of the results obtained: Two kinds of change in amplitude were observed in the normalized auto power spectral density of the neutron flux, emanating form: (a) fluctuating displacements of the boiling zone bottom boundary, and (b) fluctuating number of passing vapor voids.

Significant differences in the resulting data were observed between the runs performed with constant and with flux-related heater input in the case of step response experiment, but not in steady noise analyses. This apparent ineffectiveness of the feed-back system is due to the relatively small value of the product of the reactivity power coefficient and the reactor power (–5×10-²%Δk/k) in the present feed-back experiment.     

Cross section adjustment method based on random sampling technique

A cross section adjustment method based on the random sampling technique is proposed. In the proposed method, correlations among cross sections and core parameters are used instead of sensitivity coefficients of cross sections, which are necessary in the conventional method. The correlations are statistically estimated by the random sampling technique. The proposed method is theoretically consistent with the conventional method and provides comparable adjusted cross sections when sufficient number of random sampling is taken into account. The proposed method would be suitable for practical light water reactor (LWR) core analysis since estimation of sensitivity coefficients, which requires considerable computational cost in typical LWR problems, is not necessary. Through a benchmark problem in simple pin-cell geometry, adjusted cross sections by the present and the conventional cross section adjustment method are compared. The adjusted cross sections by the present method well reproduce the conventional ones, thus the feasibility of the present method is confirmed.     

Requisite Accuracy for Hot Spot Factors in Fast Reactors

In the thermal design of a fast reactor, it should be most effective to reduce hot spot factors to the lowest possible level compatible with safety considerations, in order to minimize the design margin for the temperature prevailing in the core. Hot spot factors account for probabilistic and statistic deviations from nominal value of fuel element temperatures, due to uncertainties in the data adopted for estimating various factors including the physical properties. Such temperature deviations necessitate the provision of correspondingly large design margins for temperatures in order to keep within permissible limits the probability of exceeding the allowable temperatures.

Evaluation of the desired accuracy for hot spot factors is performed by a method of optimization, which permits determination of the degree of accuracy that should minimize the design margins, to give realistic results with consideration given not only to sensitivity coefficients but also to the present-day uncertainty levels in the data adopted in the calculations. A concept of “degree of difficulty” is introduced for the purpose of determining the hot spot factors to be given higher priority for reduction.

Application of this method to the core of a prototype fast reactor leads to the conclusion that the hot spot factors to be given the highest priority are those relevant to the power distribution, the flow distribution, the fuel enrichment, the fuel-cladding gap conductance and the fuel thermal conductivity.     

A Method for Measuring Absolute Reactor Power through Neutron Fluctuation

A new principle is presented for obtaining absolute reactor power by processing the random fluctuation of neutron flux based on the stochastic nature of nuclear reactions. The required combination of instruments to carry out experiments is described, and experimental results obtained in a swimming pool reactor are reported. The power spectral density of the output current of an ion chamber located near the reactor core is determined by reactor kinetic parameters such as delayed neutron yield, life time, ν (mean number of neutrons generated per fission) and counter efficiency as well as by the total number of neutrons in the core, which is a measure of absolute power.

Using either logarithmic amplifier or reactivity meter, absolute reactor power can be measured without any information about detector efficiency. This method has such merits as easiness and simplicity in operation, ability to measure absolute power in the range 0.01～100 W where other methods are inapplicable, and negligible effect of changes in core configuration or in detector position.

The results of actual reactor experiments with this method proved to agree fairly well with those of absolute measurement by gold foil activation.     

Estimation of Error Propagation in Monte-Carlo Burnup Calculations

A formulation has been established to estimate the error propagation in Monte-Carlo burnup calculations. The uncertainties in cross sections and the statistical errors in Monte-Carlo calculations are considered as error sources, and the error propagation of number densities of individual nuclides over a burnup period is formulated. The present formulation is applied to the burnup calculation of a simplified fast reactor core. The components of the errors in number densities due to the statistical error are up to 0.92% even when the history number is small as 10⁴. On the other hand, the components due to the cross section error are about 2–5% for the number densities of ²³⁵U, ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu and ²⁴²Pu, and about 7.3% for the fission-product. Thus the contribution is mainly due to the cross section errors. The error propagation of the number densities due to the statistical errors at individual burnup steps is investigated by dividing the burnup period into two steps. The error propagation is not serious for the problem treated here because the component due to the statistical error is much smaller than that due to the cross section error.     

Calculation of effective delayed neutron fraction with Monte Carlo perturbation techniques

New methods are proposed to estimate the effective delayed neutron fraction β_eff in Monte Carlo calculations: the eigenvalue methods jointly used with the differential operator sampling and correlated sampling techniques. In particular, the eigenvalue method with the differential operator sampling technique has a distinct feature that it theoretically gives an exact β_eff value. To verify the proposed methods, Monte Carlo calculations are performed for several systems with simple geometry. It is found that the results obtained with the proposed methods agree with the reference deterministic results within sufficiently small statistical uncertainties. The indirect perturbed source effect must be taken into account to estimate an exact β_eff value.     

Improvement of Monte Carlo Lattice Burnup Calculation Performance with the Correlated Sampling Method

For next generation reactor designs, which are attempting wide variations of assembly configurations, the flexibility Monte Carlo method holds is attractive, but still costly for repetitive design study works. This paper presents an advanced correlated sampling (ACS) method which was developed to speed up Monte Carlo lattice burnup calculations. The ACS method is the combination of the correlated sampling method and a pseudo-scattering technique. All burnup steps are considered as consecutive perturbed problems using the same neutron collision history, which is pre-calculated based on a selected unperturbed problem. Since neutron weights can be adjusted on every collision point, rather than along paths between them, the perturbed calculation is very fast and the neutron collision history is light enough to be stored in memory or physical storage, which is an indispensable feature for consecutive perturbed calculations. The presented theory shows that the ACS method has good potential to work for a wide range of neutron absorption variations, the dominant perturbation in the lattice burnup. In an example calculation on a BWR lattice, the ACS calculation results of 600,000 neutrons/step agree well with the independent Monte Carlo runs of 20,000,000 neutrons/step within 0.1%dk/k in terms of k_? throughout 95 steps (~50GWd/t). Average calculation time of neutron tracking with the former method is 3.4 s/step with 600,000 neutron histories on a single processor of an Alpha21164-600 MHz, and the speed-up factor against the Monte Carlo calculation turns out to be about 100.     

Control of Repetitive-Accelerator Fast-Pulsed Reactor

Control algorithms are proposed for on-line computer control of the repetitive-accelerator fast-pulsed reactor.

Firstly, simplification of the reactor core dynamics is obtained by regression analysis technique in order to obtain a compact model of the pulsed reactor.

Secondly, a start-up control algorithm to avoid severe thermal shock in the fuel rod is proposed. This is essentially a programmed control, whereby the external neutron source from the accelerator and the external reactivity are controlled in such manner that the fuel temperature would follow a predetermined time trajectory. Applicability of the method to the reactor start-up is proved by a computer simulation experiment.

Thirdly, a control concept for constant power operation is proposed. This is composed of two control loops: (1) a fine control loop, which responds quickly to small variations of reactor power, and (2) a diagnosis and adaptation loop, which judges whether a large deviation from set-up value observed on the reactor power output is due to an abnormality in the external neutron source, or to a change of the set-up value itself, and then adaptively returns the reactor power to the set-up value. A computer simulation experiment demonstrated the validity of the control system characteristics.

A discussion is also presented on the problems foreseen in applying the proposed methods to the actual design of a digital computer control system.     

改进弦长抽样方法开发及在弥散燃料蒙特卡罗模拟的应用

弥散型燃料广泛应用于高温气冷堆、事故容忍燃料、实验研究堆及核动力舰船等，是重要的燃料类型之一。弦长抽样(CLS)方法可简化弥散燃料几何建模，提高计算效率，然而传统CLS方法只能描述单种颗粒的填充，同时在高体积填充率时误差较大。针对CLS方法的两大问题，本文在自主化堆用蒙特卡罗程序RMC中开发了改进CLS方法，并应用于全陶瓷微胶囊封装燃料棒算例及含毒物颗粒的高温堆燃料球算例。计算结果表明，改进CLS方法可解决多种颗粒混合填充的问题，并且可保证体积填充率的准确性，为弥散燃料的临界及燃耗计算提供了高效、精确的方法。     

Combination of sensitivity-based and random sampling-based methodologies for efficient uncertainty quantification calculations with control variates method

A combined method of the sensitivity-based and random sampling-based methodologies is proposed for efficient uncertainty quantification calculations. The proposed method is based on the control variates (CV) method, in which a mean value of a target parameter can be estimated efficiently with a help of a mockup parameter whose mean value is well known. Standard deviations can be also efficiently estimated from two mean values of stochastic parameters; a target parameter itself and its square. In the present work, the CV method is applied to a toy problem, in which a linear approximation to a target parameter is regarded as a mockup parameter. This case corresponds to our proposed method to combine the sensitivity-based and random sampling-based methodologies. Numerical results reveal that the proposed method efficiently works. As a preliminary test of application of our proposed method to realistic problems, nuclear fuel burnup calculations are considered, and uncertainties of nuclides number densities after burnup are calculated. Uncertainties of number densities of cesium-134 and europium-151 are calculated by the proposed method, and it is demonstrated that we can carry out uncertainty quantification calculations more efficiently with our proposed method than with the normal random sampling method.     

Extended cross-section adjustment method to improve the prediction accuracy of core parameters

An extended cross-section adjustment method has been developed to improve the prediction accuracy of target core parameters. The present method is on the basis of a cross-section adjustment method which minimizes the uncertainties of target core parameters under the conditions that integral experimental data are given. The present method enables us to enhance the prediction accuracy better than the conventional cross-section adjustment method by taking into account the target core parameters, as well as the extended bias factor method. In addition, it is proved that the present method is equivalent to the extended bias factor method when only one target core parameter is taken into account. The present method is implemented in an existing cross-section adjustment solver. Numerical calculations verify the derived formulation and demonstrate an applicability of an adjusted cross-section set which is specialized for the target core parameters.